Lines of Invariant Points, Invariant Lines
Invariant means something that doesn't change. A Linear Transformation will transform all the points on a grid, but invariant points will remain in their original position. We will look at invariant points, lines of invariant points, and invariant lines under linear transformations.
Prerequisites
You'll need to have an understanding of Linear Transformations.
Alternative Methods
Eigenvalues and eigenvectors may act as a more practical alternative method here, that will also work well in higher dimensions.
Invariant Points
Say we have a linear transformation represented by matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Then, an invariant point is any point which stays the same under the transformation.
In other words, the point A with position vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{x}} is invariant if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\underline{x} = \underline{x}\quad(*)} .
Since position vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{x}} is really shorthand (in 2-dimensions) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}x\\y\end{pmatrix}} , it can be clearer to write this as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}x\\y\end{pmatrix}} .
In 3-dimensions, no problem: we could say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}x\\y\\z\end{pmatrix}} .
For example, let's consider the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = \begin{pmatrix}2 &4\\1 &5\end{pmatrix}} and the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(4,-1)} .
We observe that .
Therefore, B(4,-1) is an invariant point (under the linear transformation M). The image of B is B itself.
Note that the origin (0,0) is always an invariant point - this is a property of linear transformations.
Lines of invariant points
A line of invariant points is a straight line where all the points are invariant.
The word 'invariant' applies to the individual points here - which happen to form a line.
With few exceptions[1], a line of invariant points must normally pass through the origin, so we can give it the (2 dimensional) equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx} .
By taking the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (*)} above, and substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx} , we can say: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\begin{pmatrix}x\\mx\end{pmatrix} = \begin{pmatrix}x\\mx\end{pmatrix}}
Invariant Lines
When we say invariant line, we mean that the line stays the same. But, the points on the line could move along the line. The individual points can change - there don't have to be invariant points. This is going to look like stretching out the points on the line, away from (or towards) the origin.
The point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{0},y_{0})} , on the line, will move to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{1},y_{1})} , also on the line.
Since we are talking about the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx} , we can say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{0},mx_{0})} will transform to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{1},mx_{1})} .
This gives us a method for finding invariant lines. Solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\begin{pmatrix}x_{0}\\mx_{0}\end{pmatrix} = \begin{pmatrix}x_{1}\\mx_{1}\end{pmatrix}} .
Example Question
Example: Find the two invariant lines in the linear transformation represented by T = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}3 &2\\2 & 3\end{pmatrix}} .
Since this is a linear transformation with a finite number of invariant lines, let's go ahead and assume that relevant lines include the origin. So, any such lines will be of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx} , which means we can avoid working with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +c} as an unknown. There are exceptions to this, but it makes sense to first identify lines which pass through the origin.
Now, for any point, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix} = \begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}} .
Therefore, for a point on an invariant line, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}3 &2\\2 & 3\end{pmatrix}\begin{pmatrix}x_{0}\\mx_{0}\end{pmatrix} = \begin{pmatrix}x_{1}\\mx_{1}\end{pmatrix}}
Multiplying out the matrices, we get two equations:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 3x_{0} + 2mx_{0}&=&x_{1}\qquad&(1)\\ 2x_{0} + 3mx_{0}&=&mx_{1}\qquad&(2) \end{align}}
And substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1}} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1)} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2)} :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} &2x_{0}+3mx_{0}&= & m(3x_{0}+2mx_{0})\qquad&\\ \implies&2x_{0}+3mx_{0}&= & 3mx_{0}+2m^2x_{0}\qquad&\\ \end{align} }
Notice that all the terms now contain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{0}} . Since the case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{0}=0} will represent the origin, which we are not looking for here, and is always invariant anyway, we can divide both sides by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{0}} :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \implies{}&2+3m&=& 3m+2m^2&\\ \implies{}&2m^2&=& 2&\\ \implies{}&m^2&=& 1&\\ \end{align} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \implies{}m=1 \text{ or }m=-1 }
Therefore, our invariant lines are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=-x} .
Footnotes
- ↑ Exceptions include the identity transformation - which is invariant everywhere, and shears, which are invariant for sets of parallel lines. In both cases, you can find an line of invariant points passing through the origin.